Intuition of Homology

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\cdots \longrightarrow C_{n+1} \overset{\partial_{n+1}} {\longrightarrow} C_n \overset{\partial_n}{\longrightarrow} C_{n-1} \overset{\partial_{n-1}}{\longrightarrow}C_{n-2}\longrightarrow\cdots

Homology: H_n = \text{Ker } \partial_n/ \text{Im }\partial_{n+1}

Intuition of Homology: C_n is some n- dimensional object, and \partial_n gets its boundary. \partial_{n-1}\circ \partial_{n}=0 suggests a boundary’s boundary is \emptyset. \text{Ker } \partial_{n-1} is the (n-1)- dimensional things that no more has a boundary. \text{Im }\partial_n is the (n-1)- dimensional things that is a boundary (of some n- dimensional thing). Now from this, it is still hard to visualize what H_{n-1} = \text{Ker } \partial_{n-1}/ \text{Im }\partial_n is. This suggests that homology does not have a general intuition.

Instead, let’s look at the simplest example in simplicial homology: a \Delta_2 labeled  by its three vertices, a,b,c. We have   \text{Im }\partial_2 = n(ab+bc+ca), \text{Ker }\partial_1 = n(ab+bc+ca), thus H_1 = 0. On the other hand, if we remove the face abc leaving only the skeleton, \text{Im }\partial_2=0, thus H_1 = \{n\}/\emptyset = \mathbb{Z}. We see that \text{Ker } \partial_{n-1} is like “the boundary of something”, and $latex \text{Im }\partial_n$ is like  “is this ‘something’ filled or not”? if it is filled, then we have no hole. if it is not filled, then we have a hole. This hole is captured by H_{n-1}. However, this intuition is only true for the \mathbb{Z} part of the homology (one \mathbb{Z} corresponds to one hole) and has not accounted for (e.g.) \mathbb{Z}/2\mathbb{Z} being a homology group.

In fact, finite part \mathbb{Z}/2\mathbb{Z} shows the degree of non-triviality a hole is. If it is just an ordinary hole, we can wrap it around as many times as we want so we get a \mathbb{Z}. However, if a hole is nontrivial meaning some wrapping actually deforms to identity (no wrapping at all), then we get some finite part like \mathbb{Z}/2\mathbb{Z}. An example would be \mathbb{R}P_n.

 

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